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Nonparametric Regression Estimation for Random Fields in a Fixed-Design

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We investigate the nonparametric estimation for regression in a fixed-design setting when the errors are given by a field of dependent random variables. Sufficient conditions for kernel estimators to converge uniformly are obtained. These estimators can attain the optimal rates of uniform convergence and the results apply to a large class of random fields which contains martingale-difference random fields and mixing random fields.

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  1. 1.

    K. S. Alexander R. Pyke (1986) ArticleTitleA uniform central limit theorem for set-indexed partial-sum processes with finite variance Ann. Probab. 14 582–597 Occurrence Handle0595.60027 Occurrence Handle832025

  2. 2.

    N. Altman (2000) ArticleTitleKrige, smooth, both or neither? Aust. New Zealand J Stat. 42 441–461 Occurrence Handle1016.62032 Occurrence Handle1802968 Occurrence Handle10.1111/1467-842X.00141

  3. 3.

    R. F. Bass (1985) ArticleTitleLaw of the iterated logarithm for set-indexed partial sum processes with finite variance Z. Wahrsch. verw. Gebiete. 70 591–608 Occurrence Handle0575.60034 Occurrence Handle807339 Occurrence Handle10.1007/BF00531869

  4. 4.

    A. K. Basu C. C. Y. Dorea (1979) ArticleTitleOn functional central limit theorem for stationary martingale random fields Acta. Math. Hung. 33 307–316 Occurrence Handle0431.60037 Occurrence Handle542479 Occurrence Handle10.1007/BF01902565

  5. 5.

    Biau G. Spatial kernel density estimation, To appear in Math. Meth. Stat.

  6. 6.

    Biau, G. and Cadre, B.: Nonparametric spatial prediction, To appear in Stat. Inference for Stochastic Processes.

  7. 7.

    E. Bolthausen (1982) ArticleTitleExact convergence rates in some martingale central limit theorems Ann. Probab. 10 IssueID3 672–688 Occurrence Handle0494.60020 Occurrence Handle659537

  8. 8.

    D. Bosq (1993) ArticleTitleBernstein-type large deviations inequalities for partial sums of stron mixing processes Statistics. 24 59–70 Occurrence Handle0810.60027 Occurrence Handle1238263

  9. 9.

    D. Bosq (1998) Nonparametric Statistics for Stochastic Processes-Estimation and Prediction 2nd edition Lecture Notes in Statistics. Springer Verlag New York

  10. 10.

    M. Carbon M. Hallin B. Wu (1996) ArticleTitleKernel density estimation for random fields: the L 1 theory Nonparametric Stat. 6 157–170 Occurrence Handle0872.62040

  11. 11.

    M. Carbon L. Tran B. Wu (1997) ArticleTitleKernel density estimation for random fields Stat. Probab. Lett. 36 115–125 Occurrence Handle0892.62017 Occurrence Handle1491080 Occurrence Handle10.1016/S0167-7152(97)00054-0

  12. 12.

    J. Dedecker (1998) ArticleTitleA central limit theorem for stationary random fields Probab. Theory Relat. Fields. 110 397–426 Occurrence Handle0902.60020 Occurrence Handle1616496 Occurrence Handle10.1007/s004400050153

  13. 13.

    J. Dedecker (2001) ArticleTitleExponential inequalities and functional central limit theorems for random fields ESAIM: Probab. Stat. 5 77–104 Occurrence Handle1003.60033 Occurrence Handle1875665 Occurrence Handle10.1051/ps:2001103

  14. 14.

    R. L. Dobrushin (1968) ArticleTitleThe description of a random fields by mean of conditional probabilities and condition of its regularity Th. Probab. Appl. 13 197–224 Occurrence Handle10.1137/1113026

  15. 15.

    Doukhan, P.: Mixing: Properties and Examples, Vol. 85 Lecture Notes in Statistics, Berlin, 1994.

  16. 16.

    M. El Machkouri (2002) ArticleTitleKahane–Khintchine inequalities and functional central limit theorem for stationary random fields Stoch. Proc. Appl. 120 285–299 Occurrence Handle1935128 Occurrence Handle10.1016/S0304-4149(02)00178-3

  17. 17.

    M. El Machkouri D. Volný (2003) ArticleTitleContre-exemple dans le théorème central limite fonctionnel pour les champs aléatoires réels Annales de l’IHP. 2 325–337 Occurrence Handle1014.60055

  18. 18.

    Guyon, X. (1995). Random Fields on a Network: Modeling Statistics and Applications. Springer, New York.

  19. 19.

    P. Hall J.D. Hart (1990) ArticleTitleNonparametric regression with long-range dependence Stoch. Proc. Appl. 36 339–351 Occurrence Handle0713.62048 Occurrence Handle1084984 Occurrence Handle10.1016/0304-4149(90)90100-7

  20. 20.

    M. Hallin Z. Lu L. Tran (2001) ArticleTitleDensity estimation for spatial linear processes Bernoulli. 7 657–668 Occurrence Handle1005.62034 Occurrence Handle1849373

  21. 21.

    M. Hallin Z. Lu L. Tran (2004) ArticleTitleDensity estimation for spatial processes: the L 1 theory J Multivariate Anal. 88 61–75 Occurrence Handle1032.62033 Occurrence Handle2021860 Occurrence Handle10.1016/S0047-259X(03)00060-5

  22. 22.

    Hallin, M., Lu, Z. and Tran, L.T.: Local linear spatial regression, Ann. Stat. 32 (2004) In press.

  23. 23.

    I A. Ibragimov (1962) ArticleTitleSome limit theorems for stationary processes Theory Probab. Appl. 7 349–382 Occurrence Handle0119.14204 Occurrence Handle10.1137/1107036

  24. 24.

    M. A. Krasnosel’skii Y. B. Rutickii (1961) Convex Functions and Orlicz Spaces P. Noordhoff LTD-Groningen The Netherlands

  25. 25.

    Z. Lu X. Chen (2002) ArticleTitleSpatial nonparametric regression estimation: non-isotropic case Acta Math. Appl. Sin. English series. 18 641–656 Occurrence Handle1019.62039 Occurrence Handle2012328 Occurrence Handle10.1007/s102550200067

  26. 26.

    Z. Lu X. Chen (2004) ArticleTitleSpatial kernel regression estimation: weak consistency Stat. Probab. Lett. 68 125–136 Occurrence Handle1058.62079 Occurrence Handle2066167 Occurrence Handle10.1016/j.spl.2003.08.014

  27. 27.

    D.L. McLeish (1975) ArticleTitleA maximal inequality and dependent strong laws Ann. Probab. 3 IssueID5 829–839 Occurrence Handle0353.60035 Occurrence Handle400382

  28. 28.

    B. Nahapetian (1991) Limit Theorems and Some Applications in Statistical Physics B. G. Teubner Verlagsgesellschaft Stuttgart, Leipzig Occurrence Handle0729.60015

  29. 29.

    B. Nahapetian A. N. Petrosian (1992) ArticleTitleMartingale-difference Gibbs random fields and central limit theorem Ann. Acad. Sci. Fenn. Series A-I Math. 17 105–110 Occurrence Handle0789.60043 Occurrence Handle1162153

  30. 30.

    E. Rio (1993) ArticleTitleCovariance inequalities for strongly mixing processes Ann. de l’ IHP. 29 IssueID4 587–597 Occurrence Handle0798.60027 Occurrence Handle1251142

  31. 31.

    M. Rosenblatt (1956) ArticleTitleA central limit theorem and a strong mixing condition Proc. Nat. Acad. Sci. USA. 42 43–47 Occurrence Handle0070.13804 Occurrence Handle74711 Occurrence Handle10.1073/pnas.42.1.43

  32. 32.

    R. J. Serfling (1968) ArticleTitleContributions to central limit theory for dependent variables Ann. Math. Stat. 39 IssueID4 1158–1175 Occurrence Handle0176.48004 Occurrence Handle228053

  33. 33.

    C. J. Stone (1982) ArticleTitleOptimal global rates of convergence for nonparametric regression Ann. Stat. 10 IssueID4 1043–1053

  34. 34.

    L. Tran (1990) ArticleTitleKernel density estimation on random fields J Multivariate Anal. 34 37–53 Occurrence Handle0709.62085 Occurrence Handle1062546 Occurrence Handle10.1016/0047-259X(90)90059-Q

  35. 35.

    L. Tran S. Yakowitz (1993) ArticleTitleNearest neighbor estimators for random fields J Multivariate Anal. 44 23–46 Occurrence Handle0764.62076 Occurrence Handle1208468 Occurrence Handle10.1006/jmva.1993.1002

  36. 36.

    Yao Q. (2003) Exponential inequalities for spatial processes and uniform convergence rates for density estimation H. Zhang J. Huang (Eds) Development of Modern Statistics and Related Topics– In: Celebration of Prof. Yaoting Zhang’s 70th Birthday. World Scientific Singapore 118–128

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Correspondence to Mohamed El Machkouri.

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Articlenote: In final form 24 January 2005

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Machkouri, M.E. Nonparametric Regression Estimation for Random Fields in a Fixed-Design. Stat Infer Stoch Process 10, 29–47 (2007). https://doi.org/10.1007/s11203-005-7332-6

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AMS Mathematics Subject Classification (2000):

  • 60G60
  • 62G08


  • nonparametric regression estimation
  • kernel estimators
  • strong consistency
  • fixed-design
  • exponential inequalities
  • martingale difference random fields
  • mixing
  • Orlicz spaces